3 ms·
So the outside of a sphere is convex (positive curvature), a saddle is concave (negative curvature)... and the inside of a sphere (AKA a concave mirror/lens) is
by jimbo1qaz 7y ago
So the outside of a sphere is convex (positive curvature), a saddle is concave (negative curvature)... and the inside of a sphere (AKA a concave mirror/lens) is also convex (positive curvature)?
- BoiledCabbage 7y agoI believe so yes. The curvature of a surface is independent of which "side" you are measuring from.
- giomasce 7y agoNo, you're wrong. In general there is not even a concept of "side", but even if there is a positively-curved surface or manifold seen from the other side is still positively-curved. And the same with negative. You can probably convince yourself of this if you think at the triangles mentioned in the article. Their angles do not change even when you change side.
- pdonis 7y agoThere is no "outside" or "inside" when you're talking about Gaussian (intrinsic) curvature. "Outside" or "inside" are about how the object is embedded in space, but Gaussian curvature is independent of how (or even if) the object is embedded in space.
- jimbo1qaz 7y agoSo Gaussian curvature reuses the words "convex" and "concave" to mean things entirely different from the everyday/lens usage?
- pdonis 7y ago> So Gaussian curvature reuses the words "convex" and "concave" to mean things entirely different from the everyday/lens usage? The everyday/lens usage is talking about how the surface is embedded in a higher dimensional space. Gaussian curvature is not. So you should not expect the meanings in the two cases to be the same.